You are starting a race on the beach at point A to reach a buoy at...

 

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You are starting a race on the beach at point A to reach a buoy at point B in the ocean, as pictured below (bird's eye view). A=(0,0) x Beach Ocean B = (b1, b2) You can run along the beach twice as fast as you can swim in the ocean. The aim is to find that the optimal location x to enter the ocean is such that 0 = 30°. (a) Let v₁ = running speed and v₂ = swimming speed. Show that the total travel time between A and B is (explain your working): 1 T(x) = VI x+ 1 v₂ √(b₁ − x)² + b². (b) Calculate T'(x). (c) Show that if v₁ = 2v2, the optimal angle to enter the ocean is 0 = 30°. Hint: Note that T(x) has no maximum (you can take arbitrary long detours) and that sin 0 (b₁-x) √(bi-x)²+b² You are starting a race on the beach at point A to reach a buoy at point B in the ocean, as pictured below (bird's eye view). A=(0,0) x Beach Ocean B = (b1, b2) You can run along the beach twice as fast as you can swim in the ocean. The aim is to find that the optimal location x to enter the ocean is such that 0 = 30°. (a) Let v₁ = running speed and v₂ = swimming speed. Show that the total travel time between A and B is (explain your working): 1 T(x) = VI x+ 1 v₂ √(b₁ − x)² + b². (b) Calculate T'(x). (c) Show that if v₁ = 2v2, the optimal angle to enter the ocean is 0 = 30°. Hint: Note that T(x) has no maximum (you can take arbitrary long detours) and that sin 0 (b₁-x) √(bi-x)²+b²